Which maths should I start with?

[Technik]

Hello all,

I've been reading a lot about cosmology lately. Mostly higher level abstractions about how things work without really wanting to get dragged or pulled down by some of the maths. My eyes kind of glaze over an equation when I see one on a wikipedia article.

But that got me to thinking; I was never particularly bad at math, even though I was always afraid I didn't understand it "all the way." I'm better with programming, which is similar, but not always as number oriented. Anyways, the last math I took was Pre-calculus Algebra, and that was about a year or so ago.

I'm not really interested in pursuing a physics degree or anything career-oriented/professional, but I would like to self-study physics and math for personal enrichment.

Any suggestions about where to start? My precalc is probably rather rusty; should I work on that? Perhaps pull that old book out and start chugging through the chapters and exercises again? Ultimately I'd like to learn calculus and trig, perhaps use that knowledge to help in programming basic orbital calculations, etc.

 

[SolsticeFire]

If your end goal is to understand the field of cosmology and 'higher level abstractions' as you put it, I'd say you ought to take the route of Classical Mechanics --> Thermodynamics --> Quantum Mechanics --> Relativity. Maybe do QFT after all that and graduate level particle physics. I don't recommend learning mathematics separately as you'll end up learning a lot of things that you wouldn't need to understand cosmology related articles on wikipedia. You can learn mathematics side by side as they come up in the physics textbooks you are tackling.

I can recommend a few books :)

Classical Mech: Taylor/Morin. (Learn Calculus along the way using Spivak, Kline, Lang. Learn Diffy Q using Tenenbaum/Arnold (Not too soon with this one) )
Thermo : Reif/Callen/ Fermi. (Still need calculus for this)
QM : Griffiths/Shankar/Sakurai (Pick up Linear Algebra Axler/Meyer/Poole)
Relativity: Wald/Stewart (Learn Diff Geometry )

Although it seems like a quick little sequence from mechanics to general relativity and QFT; I assure you that it will take a lot of time and effort to finally understand those higher abstractions. But hey, you won't have to just glaze over those pesky equations anymore! You'll know exactly what they mean and the underlying substance beneath them. :)

Good Luck

SolsticeFire

PS: Once you get to a high enough level, you'd have to know a lot more esoteric mathematics than you'd like to (but hopefully you'd have a deep appreciation for mathematics by then! :P), but you can slowly pick them up as they come along. As a result, I have left out a lot of mathematics that you'd need to know eventually.

 

[Astrum]

If your end goal is to understand the field of cosmology and 'higher level abstractions' as you put it, I'd say you ought to take the route of Classical Mechanics --> Thermodynamics --> Quantum Mechanics --> Relativity. Maybe do QFT after all that and graduate level particle physics. I don't recommend learning mathematics separately as you'll end up learning a lot of things that you wouldn't need to understand cosmology related articles on wikipedia. You can learn mathematics side by side as they come up in the physics textbooks you are tackling.

I can recommend a few books :)

Classical Mech: Taylor/Morin. (Learn Calculus along the way using Spivak, Kline, Lang. Learn Diffy Q using Tenenbaum/Arnold (Not too soon with this one) )
Thermo : Reif/Callen/ Fermi. (Still need calculus for this)
QM : Griffiths/Shankar/Sakurai (Pick up Linear Algebra Axler/Meyer/Poole)
Relativity: Wald/Stewart (Learn Diff Geometry )

Although it seems like a quick little sequence from mechanics to general relativity and QFT; I assure you that it will take a lot of time and effort to finally understand those higher abstractions. But hey, you won't have to just glaze over those pesky equations anymore! You'll know exactly what they mean and the underlying substance beneath them. :)

Good Luck

SolsticeFire

PS: Once you get to a high enough level, you'd have to know a lot more esoteric mathematics than you'd like to (but hopefully you'd have a deep appreciation for mathematics by then! :P), but you can slowly pick them up as they come along. As a result, I have left out a lot of mathematics that you'd need to know eventually.

I don't completely agree with this. You need a fair amount of calculus to be able to go through Taylor's Mechanics.

My advice would be to review Pre-calculus, then go into calculus. If you're goal is only to understand math in relation to physics (ie. not like a mathematician), you could go with Kline's "Calculus: An Intuitive and Physical Approach".

And I also don't recommend going straight into second year university physics classes. You probably should start with an intro physics text first.

 

[Technik]

If your end goal is to understand the field of cosmology and 'higher level abstractions' as you put it, I'd say you ought to take the route of Classical Mechanics --> Thermodynamics --> Quantum Mechanics --> Relativity. Maybe do QFT after all that and graduate level particle physics. I don't recommend learning mathematics separately as you'll end up learning a lot of things that you wouldn't need to understand cosmology related articles on wikipedia. You can learn mathematics side by side as they come up in the physics textbooks you are tackling.

I can recommend a few books :)

Classical Mech: Taylor/Morin. (Learn Calculus along the way using Spivak, Kline, Lang. Learn Diffy Q using Tenenbaum/Arnold (Not too soon with this one) )
Thermo : Reif/Callen/ Fermi. (Still need calculus for this)
QM : Griffiths/Shankar/Sakurai (Pick up Linear Algebra Axler/Meyer/Poole)
Relativity: Wald/Stewart (Learn Diff Geometry )

Although it seems like a quick little sequence from mechanics to general relativity and QFT; I assure you that it will take a lot of time and effort to finally understand those higher abstractions. But hey, you won't have to just glaze over those pesky equations anymore! You'll know exactly what they mean and the underlying substance beneath them. :)

Good Luck

SolsticeFire

PS: Once you get to a high enough level, you'd have to know a lot more esoteric mathematics than you'd like to (but hopefully you'd have a deep appreciation for mathematics by then! :P), but you can slowly pick them up as they come along. As a result, I have left out a lot of mathematics that you'd need to know eventually.

I don't completely agree with this. You need a fair amount of calculus to be able to go through Taylor's Mechanics.

My advice would be to review Pre-calculus, then go into calculus. If you're goal is only to understand math in relation to physics (ie. not like a mathematician), you could go with Kline's "Calculus: An Intuitive and Physical Approach".

And I also don't recommend going straight into second year university physics classes. You probably should start with an intro physics text first.

Thanks for the suggestions. It's good to hear I can learn the math as it's required while going through physics. I'm not really looking to study as a mathematician or any of that. Also, I'm mostly interested in classical mechanics and relativity.

What would be the best learning resource for that? Either for physics or the required maths. I take it Kline's Calculus would be great for the maths. Any suggestions for a place to get started in physics?

 

[halo31]

Depending on how far you would want to go you still need quite a bit of math. A good understanding of Calculus 1-3, linear algebra, and differential equations is a must. Then from there a good understanding of differential geometry, manifolds, and knot theory is necessary as well as a flavor of abstract algebra. Just look at it this way a graduate student in physics takes the same amount of math as you would if you were to pursue a bachelor's degree. Maybe even more. I'm no physicist but that's my understanding. If your just looking to learn lower division physics then you would just need to know Calculus 1-3, linear algebra, and differential equations. And there are plenty of good books on these things and shouldn't be difficult to learn on your own.

 

[Astrum]

Thanks for the suggestions. It's good to hear I can learn the math as it's required while going through physics. I'm not really looking to study as a mathematician or any of that. Also, I'm mostly interested in classical mechanics and relativity.

What would be the best learning resource for that? Either for physics or the required maths. I take it Kline's Calculus would be great for the maths. Any suggestions for a place to get started in physics?

Kline doesn't cover vector calculus, nor does he cover infinite series and sequences very well. You're going to need another book for those.

As far as intro physics goes, I like Halliday and Resnick, although it's watered down a lot. I'd suggest this first, go through the important chapters. So, that means most of the mechanics chapters, plus E&M and relativity.

After that you could go on to Mechanics by Taylor. I don't think there's a huge amount of material to SR (could be wrong), so that shouldn't be too bad.

 

[KClose1983]

I've been putting together my own kind of self-study course myself as well. I found Kim Seward's (West Texas A&M University) tutorials to be a great reference for College Algebra:

http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/index.htm

I don't know where you're looking to start, but if you don't know where to start yourself, this is a great site to check how much you remember of basic algebra before moving on to more advanced mathematics - it covers things like quadratic equations, binomial theorem, logarithms, complex numbers, etc.

If, after checking this out, you find all of this to be already familiar, you might try MIT's Open Courseware for Physics video lectures as well as Calculus of a Single Variable video lectures. I just started watching the calculus videos recently, but they are very helpful. You can find the calculus MIT intro to Calculus of a single variable video here:

http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/

I have trouble with wikipedia sometimes, primarily due to the way the information is organized and lack of examples in the presentation of abstract material. However, these resources coupled with wikipedia as well as some books I have picked up at the library have made for a great pool for cross-referencing.

For physics, depending upon where you wish to start, I have found the following two resources to be very helpful for both brushing up on older material as well as introducing more challenging and mathematically demanding material as well:

http://www.physicsclassroom.com/

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

 

[KClose1983]

Also, you can find Einstein's work here:

http://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory